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Questions tagged [topology]

In mathematics, topology examines the properties of space (such as connectedness and compactness) that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing.

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Fluxes on a finite group $G$

So I've been studying about topological quantum computation and I have a few questions I haven't been able to solve. The first one is why fluxes take values on a finite group $G$? Does it have to do ...
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3answers
117 views

Rigorous procedure of gluing together two spacetimes

There seems to exist a procedure of "gluing two spacetimes together". In particular I've seem this mentioned in the context of gravitational collapse. The examples I've seem where that of gluing ...
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0answers
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What is the shape of the universe? [duplicate]

If it's flat then how a volume can be flat? And I've read that it's actually not flat .....it's a "Poincaré dodecahedral space". Any suggestions for books posts or articles are highly appreciated.
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Does the Einstein field equation uniquely determines the topology of spacetime? [duplicate]

I am trying to understand whether the Einstein field equation uniquely determines the topology of spacetime. As far as I know, given a metric, we can always find the induced topology. However, I was ...
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String topology in string theory

How do string topology, string field theory and topological strings interact? Does anybody see a global picture? By string topology I mean the TQFT based on the homology of the space of loops ...
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31 views

Can the shape of our Universe be a Mobius strip? [duplicate]

The Friedmann Equations describe three possibilities for the shape of our using General Relativity, I read in a book that the shape of our Universe is a 3-sphere such that in any direction if you ...
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32 views

Critical points of vector field with zeros in the magnitude

I am studying a vector field which has critical points (sources, sinks, saddle points and centers). The magnitude of the vector field goes to zero smoothly in these points, however. Contrast that to ...
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1answer
71 views

Berry phase: Spin in a magnetic field parameter space manifold

Canonical example for Abelian Bery phase is a spin in a magnetic filed, e.g.. Usually authors calculate spin eigenstates, conclude that they don't depend on B in spherical components and so deduce ...
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1answer
55 views

Topological theta-term as a background electric/magnetic field?

The topological $\theta$-term in the Schwinger model (1+1-dimensional QED) can be interpreted as a background electric field, as explained in Chapter 7.1.2 of Tong's lecture notes. The same holds true ...
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1answer
58 views

Theta-dependence of massive Schwinger model

I've read in Coleman's paper on the massive Schwinger model (and in other papers on the same topic, like this one) that the model's Hamiltonian contains a topological $\theta$-term. However, if I ...
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1answer
104 views

Black holes in p-adic gravity/ultra-metric metric field? [closed]

As a radically different to beyond standard general relativity, at least from the type of geometry it deals with: Consider p-adic gravity and/or general relativity defined on certain ultra-metric ...
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Geometry of the Earth [migrated]

Is this true that every simple closed curve on the earth can be deformed continuously to a point without leaving the earth? Is the earth compact? Now if we consider the earth as a 2-manifold, can we ...
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What’s the topology of critical region?

Duhem said the aim of physics is natural classification. I think topology and geometry are a wonderful way to link analogous parts among different phenomena. Thus we can classify and predict facts. ...
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1answer
171 views

“Hidden” theta-term in Hamiltonian formulation of Yang-Mills theory

I've read in David Tong's lecture notes on gauge theory that the Hamiltonian of Yang-Mills theory does not depend on the angular parameter $\theta$, because it can be absorbed in the electric field: $...
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32 views

Symmetry operators of a Bloch Hamiltonian

Consider a lattice with a 3 atom basis, e.g. the Lieb lattice, and some completely arbitrary on-site energies and hopping energies and phases between the different atoms. In momentum space we can ...
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1answer
38 views

Change in area on a 3-sphere bounded by a trajectory due to a differential change in trajectory

I have a 3 dimensional spherical topology, and I draw a curve onto the sphere labelled by $\vec{n}(\vec{r},t)$. The area bounded by the curve is termed the "Wess Zumino Action" (Hence my motivation to ...
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47 views

Hausdorff property in Minkowski spacetime

In the 4-dimensional Minkowski spacetime, for a given point $x = (x^0,x^1,x^2,x^3)$, its timelike future/past set is defined as, $$ I^{\pm}(x) = \{y =(y^0,...,y^3) \in \mathbb{R}^4 : \eta_{\mu \nu}(y-...
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1answer
114 views

Why doesn't the $\theta$ Angle Renormalize?

The $\theta$ term for Yang-Mills takes the form $$L_{\theta}=\frac{\theta}{64\pi^2}\varepsilon^{\mu\nu\rho\sigma}F^a_{{\mu\nu}}F^a_{\rho\sigma}$$ A fact that I have heard is that $\theta$ does not ...
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21 views

Inside a shell full of strings, what are the components of the network, and what the possible defects?

Imagine that we have a (very) large shell full of holes, and flexible strings go (randomly) from one hole to another. Inside the shell, there will be a network of strings. Can one split this network ...
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1answer
32 views

Weyl Hamiltonian - Monopole in momentum space

Consider the Weyl Hamiltonian in momentum space: $$ \mathcal{H} = \hbar v_F \chi (\bf{\sigma}\cdot{\bf k}) , $$ where $\sigma^i$ are the Pauli matrices and $\chi$ the chirality of the Weyl node. ...
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1answer
42 views

Classical field theory with fields on different base spaces

Keeping things at a "basic level", a field is a function from a base manifold (of dimension D) to some other space. Usually the base manifold is the spacetime but may be something different (a lattice,...
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2answers
94 views

Why are systems in classical mechanics sets and not vector spaces?

One of my professors told me that a state in a classical system is an element of a set, while in quantum it's an element of a vector space. From that, we combine systems in classical physics using a ...
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2answers
66 views

Triangle inequality and the pseudo-metric of general relativity

The pseudo-metric, such as those used in general relativity, can be expressed as the following sum: $$ ds^2=\sum_{\nu}\sum_{mu}g_{\nu\mu}dX_\nu dX_{\mu} $$ The elements $g_{\nu\mu}$ can be organized ...
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1answer
62 views

Parameter space of $SO(3)$ and $SU(2)$

Is it parameter space of $SO(3)$ and $SU(2)$ are same? can we use quaternions to represent both groups? what about their connectedness?
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1answer
151 views

Basics of topological order and its relation to entanglement

What is a topological order that drives a topological phase transition? How is it different from say magnetic ordering or the superfluid ordering? What is its relation with entanglement? Please ...
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0answers
34 views

IR divergence Feynman diagram topology query

I am trying to calculate the superficial degree of Infrared divergence. To do this I am reading section 12 of this source. It seems you can calculate it by a method involving the 'shrinking' of ...
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0answers
16 views

What does the interferogram in Michelson interferometer mean about the topology of the specimen?

I'm want to know how can I evaluate the surface of a fibre by using the interferometer of Michelson. The object beam in this method is reflected from the object and combined with the reference beam to ...
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0answers
97 views

Isn't mass from pure topology in the absence of matter a contradiction?

Consider the 3 dimensional projective space minus the infinity point, and empty of matter. We see that it has ADM mass. In other words, a perfectly fine geometry, orientable and asymptomatically flat,...
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1answer
39 views

Can any body suggest me a set up that I can build it up to study the surface roughness of transparent polymeric fibres? [closed]

I need to study the surface of nylon-6 blank fibres with a thickness of nearly 40 micrometre. I work in optics lab with many optical components such as beam splitters, lenses, mirrors, etc; can I ...
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0answers
15 views

Can I build up a profilometry set up, to study the surface roughness of polymer fibres?

I need to study the surface roughness of some fibres, I was told that the best way is the profilometer, but can I design and build up a set up of a profilometer?
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0answers
10 views

How interferometric techniques can be used to investigate the topology of polymers?

I need to study the topography of a specific polymer, knowing that my lab is an optics lab with many interferometric techniques, how can I use these techniques to investigate the topology?
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Topological shape of the equilibrium point

"All dynamical system possess topological shapes that characteristics it's equilibrium point"-so my question is what is the topological shape of the equilibrium point for a cart and Inverted pendulum ...
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1answer
130 views

What is the atomic limit?

I am attempting to grasp topological superconductivity for an assignment and in trying to understand what makes a quantum system topological have came across the following paragraph; "In the case ...
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1answer
62 views

Chart(s) of space-time as a smooth manifold

So we all know that space-time in general relativity is modeled as a smooth (pseudoRiemannian) manifold. Each point (event) on space-time is labeled with a unique coordinate $(t,x,y,z)$ in a specific ...
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33 views

coulomb interaction on a ring

My qualitative understanding is that the mathematical form of the interaction between particles is constrained by their gauge symmetry, so that, for example, the U(1) gauge symmetry in QED gives rise ...
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4answers
357 views

Weyl spinor representations and the Lorentz group

I'm currently trying to read up on the Lorentz-group and it's representations. I've found a couple of posts here on stack-exchange that I find helpful and confusing at the same time, so I would be ...
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1answer
64 views

Physical meaning of theorem

This is the image of theorem from V.I Arnold's Mathematical method of mechanics. I understood the example given in text. But I want to know what is physical meaning of example? Can anybody help?
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Which mechanical system has a Moebius strip as its configuration space?

A harmonic oscillator has a line as its configuration space. A pendulum has a circle as its configuration space. Which system has a Moebius strip as its configuration space?
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0answers
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The connection between symmetry and classifying spaces of a group

I recently read the following statement: "For any type of mathematical object, an object of that type with $G$ symmetry “is” a map from [its classifying space] $BG$ to the space of all objects ...
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1answer
93 views

Does the vierbein contain any extra information?

The vierbein from General relativity has $D(D+1)/2$ independent components when accounting for the $O(3,1)$ gauge symmetry. The metric has the same degrees of freedom. But does the vierbein contain ...
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1answer
101 views

Does a geodesic always extremize its path length? [duplicate]

I've learned that a geodesic maximizes its proper time in Minkowski spacetime. Is this still true in general curved spacetime? In other words, does the geodesic equation give the globally extremal ...
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0answers
54 views

What does it mean to define a spin-structure on a manifold? [closed]

I'm trying to think about what information I need to add to a manifold that it describes a spin structure? I know you can have spin-structure on a 2d plane, a 2-sphere. I also know you can define a ...
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0answers
45 views

The Simply Connected Manifold for $SU(3)$

$U(1)$ is the 1-sphere (S^1); $SU(2)$ is the 3-sphere (S^3); $SU(3)$ is _______________ (fill in the blank). What simply connected manifold is $SU(3)$ (isomorphic to)?
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1answer
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Space/time movement in [closed]

Does time move in a seamless orientation, or can there be rips or tears in time that we do not know about? I'am asking out of curiosity only, not a study problem.
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1answer
64 views

The sphere $S^d$ is Euclidean space $E^d$ with infinity identified as a single point

I'm reading about anti de Sitter spacetime, and I found the following statement: $$ds^2 = \frac{1}{\cos^2 \psi} \big( -dt^2 + d\psi^2+ \sin^2 \psi d\Omega_{d-2}^2 \big).$$ Thus, the spatial ...
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0answers
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Pure math courses for physicists: Topology [closed]

I'm in my bachelor in physics. In a couple of weeks I start my last year, and I'm interested in taking some pure math courses. As you see, I like the theoretical point of view, but I don't know if the ...
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0answers
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A good instruction on Symmetry enriched Topological phases

I am looking for a good introduction to SETs, and topologically ordered phaeses it should be something describing first principles and gives a good explanation on the basics and the logic of this ...
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1answer
139 views

Reflection Vector (Ray Tracing)

Snell's law of refraction at the interface between 2 isotropic media is given by the equation: \begin{align} \tag{1} n_1 \,\text{sin} \,\theta_1 = n_2 \, \text{sin}\,\theta_2 \end{align} $\qquad$ ...